Given an infinite finitely generated group \(G\)  with finite generated set \(S\)  one can construct the Cayley graph \(\Gamma (G,S)\) , which is an infinite regular graph, and consider the Laplacian on it.
Alternatively, one may also consider the adjoint or the Markov operator or more generally a convolution operator on \(l^2(G)\)  given by an element in the group algebra \(\mathbb {C}G\) .
Information about the group can be extracted from properties of these spectral measures; for instance, Kesten's Theorem asserts that \(G\)  is amenable if and only if the spectral radius of the Markov operator is 1.
In [1]}, Kesten initiated the study of random walks on groups and their spectral properties.
He studied the Markov operator on free groups and found their spectral measures, which are all absolutely continuous.
